Math 151
Section 1.1
Vectors
A vector is a description of how to get from one point to another using both a direction and a distance or magnitude . We can describe a vector by using an ordered pair
a 1
and a
2
are the components of the vector. a = a
1
, a
2
Example: Sketch the vector with initial point A (3, 4) and terminal point B (4, − 2).
where
What are the components of AB ?
Definition: Given points
(
1
, y
1
)
and
(
2
, y
2
)
, the vector a corresponding to AB is a = x
2
!
x
1
, y
2
!
y
1
Definition: The magnitude (or length, distance) of vector a is denoted by | a | and is calculated by using the distance formula. a = a
1
2 + a
2
2

Example: Find the components of the vector r given that:
A.
| r | = 2 and r makes an angle of 60 ° with the positive x -axis.
B. | r | = 7 and r makes an angle of − 45 ° with the positive x -axis.
Operations on Vectors Let a = a
1
, a
2
and b = b
1
, b
2
.
Vector Sum a + b = a
1
+ b
1
, a
2
+ b
2
Vector Difference a !
b = a
1
!
b
1
, a
2
!
b
2
Multiplication by a scalar c c a = ca
1
, ca
2
Properties of Vectors Let a , b , and c represent vectors, scalars.
0 = 0,0 , and m and n represent
1.
5.
2.
3.
4.
a a + b + c
a a
+
+
+
( b
0
=
=
( ) b a
+
=
) a
=
0
( a + b
)
+ c
6.
7.
8.
( m
( a m + n
( )
+ a
1 a = a
) b a
=
)
=
= m m m n a a a
+
+ m n a b
Example: Let a = !
1,2 and b = 4,3 .
A.
3 a + 4 b
B. a !
b
Math 151

Math 151
Unit Vectors A unit vector is a vector with magnitude 1. To determine a unit vector, u, in the direction of a , divide by the magnitude of a , u = a
. a
Standard Basis Vectors Unit vectors in the direction of the positive x - and y -axes with magnitude 1 are called the standard basis vectors and are represented by i = 1,0 and j = 0,1 . The standard basis vectors can be used to represent all other vectors.
Example: Let a = !
1,2 and b = 4,3 .
A. Find a unit vector in the direction of b .
B. Find a vector with magnitude 3 in the direction of b .
Example: Let m a + n b = c . a = 1,5 , b = 3, !
1 , and c = 8,6 . Find the value of scalars m and n such that

Math 151
Applications to Physics and Engineering A force is represented by a vector because it has both magnitude (measured in pounds or newtons, N) and direction. If several forces are acting on an object, the resultant force experienced by the object is the vector sum of the forces.
Example: The ship’s captain walks due west on the deck of the ship at 3 mph. The ship is moving north at 22 mph. Find the speed and direction of the captain relative to the surface of the water.
Example: Two forces, S and T , are acting on an object at a point P as shown. | S | = 20 pounds and measures a reference angle of 45 ° . | T | = 16 pounds and measures a reference angle of 30 ° . Find the resultant force and its magnitude and direction.

Math 151
Example: Suppose that a wind is blowing from the direction N45 ° W at a speed of 50 km/hr. A pilot is steering a plane in the direction N60 ° E at an airspeed (speed in still air) of 250 km/hr.
Find the true course (direction of the resultant velocity vectors of the plane and wind) and ground speed (magnitude of resultant).